Enhanced thickness calibration and shading correction for automatic X-ray inspection

• US 6,201,850 B1
• Filed: 01/26/1999
• Issued: 03/13/2001
• Est. Priority Date: 01/26/1999
• Status: Expired due to Fees

• Current Assignee: Agilent Technologies, Inc.
• Original Assignee: Agilent Technologies, Inc.
• Inventors: Heumann, John M.
• Primary Examiner(s): BRUCE, DAVID VERNON
• Application Number:
  US09/237,401
• Time in Patent Office: 777 Days
• Field of Search: 378/22, 378/54, 378/56, 378/58
• US Class Current: 378/56
• CPC Class Codes: G01B 15/025 by measuring absorption
• CPC Class Codes: G01N 23/044 using laminography or tomosynthesis

Abstract

An X-ray inspection system incorporates an improved technique for determining, in an X-ray image of a multilayered assembly, the gray level component of a first material in the presence of a second material. The total gray level of the image is dependent upon the physical characteristics of each material comprising the assembly. The present invention accurately determines the component of the total image gray level due to the first material. In the case of circuit board inspections using X-ray images of solder connections, a calibration procedure facilitates the direct conversion of the gray level component due to the solder connection to the thickness of the solder connection.

Claims

1. A method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said method comprising the steps of:

providing a calibration standard having;

a) multiple combinations of a first known thickness of the first absorbing material (denoted by t_M1,1) in combination with three thicknesses of the second absorbing material (denoted by t_M2,1, t_M2,2 and t_M2,3); and

b) multiple combinations of a second known thickness of the first absorbing material (denoted by t_M1,2) in combination with three thicknesses of the second absorbing material (denoted by t_M2,4, t_M2,5 and t_M2,6);

determining the values of first, second and third foreground parameters (denoted by F₁, F₂ and F₃) wherein;

a) the first foreground parameter F₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,1;

b) the second foreground parameter F₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,2; and

c) the third foreground parameter F₃ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,3;

determining the values of first, second and third background parameters (denoted by B₁, B₂ and B₃) wherein;

a) the first background parameter B₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,1;

b) the second background parameter B₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,2; and

c) the third background parameter B₃ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,3;

determining the values of fourth, fifth and sixth foreground parameters (denoted by F₄, F₅ and F₆) wherein;

a) the fourth foreground parameter F₄ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,2 in combination with the second absorbing material having the thickness t_M2,4;

b) the fifth foreground parameter F₅ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,2 in combination with the second absorbing material having the thickness t_M2,5; and

c) the sixth foreground parameter F₆ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,2 in combination with the second absorbing material having the thickness t_M2,6;

determining the values of fourth, fifth and sixth background parameters (denoted by B₄, B₅ and B₆) wherein;

a) the fourth background parameter B₄ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,4;

b) the fifth background parameter B₅ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,5; and

c) the sixth background parameter B₆ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,6; and

determining a first functional form of a non-linear function, y₁(x), which describes the value of the foreground minus the background (y₁=F−

B) as a function of background (x=B) such that the non-linear functional form;

a) approximates the following values of foreground minus background;

(F₁−

B₁), (F₂−

B₂), (F₃−

B₃), (F₄−

B₄), (F₅−

B₅) and (F₆−

B₆;

b) supports extrapolation beyond the range of the values of foreground minus background {(F₁−

B₁), (F₂−

B₂), (F₃−

B₃), (F₄−

B₄), (F₅−

B₅), (F₆−

B₆)} and/or foreground {F₁, F₂, F₃, F₄, F₅, F₆} and/or background {B₁, B₂, B₃, B₄, B₅, B₆}; and

c) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system.

2. The method of claim 1 wherein the steps of determining the values of the foreground and background parameters further comprise the steps of:

illuminating the calibration standard with a beam of X-rays having the incident X-ray beam intensity, wherein the beam of X-rays is produced by an X-ray source; and

measuring the values of the foreground and background parameters with an X-ray detector.

3. The method of claim 1 wherein the steps of determining the values of the foreground and background parameters further comprises the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors:

a) spectral characteristics of the X-ray source; and

/or b) angular distribution of X-rays produced by the X-ray source; and

/or c) stopping power and spectral sensitivity of the X-ray detector; and

/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength.

4. The method of claim 1 wherein the foreground parameters F_i are described by a functional form, y_F:

5. The method of claim 1 wherein the step of determining a first functional form of a smoothly varying non-linear function which expresses the value of the foreground minus the background (y₁=F−

B) as a function of background (x=B) comprises the step of selecting a function of the form;

6. The method of claim 1 further comprising the steps of:

selecting a reference background level (x=B_R);

determining the values of foreground minus background (F_Ri−

B_Ri) at the reference background level (B_R) for multiple known thicknesses of the calibration standard using the smoothly varying non-linear function y₁ which expresses the value of the foreground minus the background (y₁=F−

B) as a function of background (x=B); and

determining a second functional form y² which expresses the values of foreground minus background (F_Ri−

B_Ri) at the reference background level (B_R) for the multiple known thicknesses of the first absorbing material as a function of the thickness of the first absorbing material.

7. The method of claim 6 wherein the step of determining a second functional form y² further comprises the step of selecting a function which is a sum of exponentials of the form:

8. The method of claim 6 further comprising the step of producing a lookup table for values of (background) vs. (foreground minus background) vs. (thickness) for one or both of the first and/or second absorbing materials.

9. The method of claim 8 further comprising the steps of:

determining the value of a seventh foreground parameter (denoted by F₇) which is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having an unknown thickness t_M1,7 in combination with the second absorbing material having an unknown thickness t_M2,7;

determining the value of a seventh background parameter (denoted by B₇) which is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the second absorbing material having an unknown thickness t_M2,7; and

using the lookup table and the values of F₇ and B₇ to determine one or both of the unknown thickness(es) of the first absorbing material (t_M1,7) and/or the second absorbing material (t_M2,7).

10. The method of claim 8 further comprising the step of interpolating between values in the lookup table.

11. The method of claim 10 where the step of interpolating further comprises the step of bilinear interpolation.

12. The method of claim 1 further comprising the step of selecting the thicknesses of the second absorbing material (t_M2,i) such that at least one of the values of the first, second and third background parameters (denoted by B₁, B₂ and B₃) is equal to at least one of the values of the fourth, fifth and sixth background parameters (denoted by B₄, B₅ and B₆).

13. The method of claim 1 further comprising the step of selecting the thicknesses of the second absorbing material (t_M2,i) such that at least two of the values of the first, second and third background parameters (denoted by B₁, B₂ and B₃) are equal and/or at least two of the values of the fourth, fifth and sixth background parameters (denoted by B₄, B₅ and B₆) are equal.

14. A method for measuring the thickness of a first material in the presence of a second material comprising the steps of:

providing a calibration standard having;

a) multiple combinations of a first known thickness of the first material in combination with a range of thicknesses of the second material; and

b) multiple combinations of a second known thickness of the first material in combination with a range of thicknesses of the second material;

exposing the calibration standard to a source of transmissive energy having an incident intensity;

detecting the intensity of the transmissive energy which passes through the calibration standard, said detecting step further comprising the step of;

acquiring multiple pairs of image data which are representative of a portion of the transmissive energy which is measured after transmission through the first and second materials, where a foreground value (F) in each pair of image data corresponds to a portion of the incident intensity which is transmitted through the known thickness of the first material in combination with one of the multiple thicknesses of the second material, and a background value (B) in each pair of transmitted intensities corresponds to a portion of the incident intensity which is transmitted through only the corresponding thickness of the second material which was in combination with the first material when the foreground value (F) was acquired;

determining fitting constants a,b and c for each member of a family of hyperbolic curves which describe delta gray values (y₁=Δ

G=F−

B) as a function of background values (B), where each curve in the family represents delta gray values for a fixed known thickness of the first material in combination with a range of thicknesses of the second material, each of the hyperbolic curves having the general form of;

15. A method for measuring the thickness of a first material in the presence of a second material comprising the steps of:

providing a calibration standard having;

a) multiple combinations of a first known thickness (t_M1,1) of the first material in combination with a range of thicknesses (t_M2,a, t_M2,b, . . . , t_M2,n1) of the second material; and

b) multiple combinations of a second known thickness (t_M1,2) of the first material in combination with a range of thicknesses (t_M2,n1+1, t_M2,n1+2, . . . , t_M2,n1+n2) of the second material;

exposing the calibration standard to a source of transmissive energy having an incident intensity;

detecting the intensity of the transmissive energy which passes through the calibration standard and determining therefrom image data which are representative of a portion of the transmissive energy which is measured after transmission through the first and second materials, said detecting step further comprising the step of;

acquiring multiple pairs of image data, where each pair includes a foreground value and a background value, for each known thickness of the first material (t_M1,1, t_M1,2) in combination with multiple thicknesses (t_M2,a, t_M2,b, etc.) of the second material;

where the foreground value (y_f) in each pair of image data corresponds to a portion of the incident intensity which is measured after transmission through the known thickness of the first material in combination with one of the multiple thicknesses of the second material, and the background value (y_b) in each pair of image data corresponds to a portion of the incident intensity which is measured after transmission through the corresponding thickness of the second material which was in combination with the first material when the foreground value (y_F) was acquired;

determining fitting constants y₀, α

_i and β

₁ from the calibration standard background values for a functional form which approximates the measured background values (y_b) as a function of the thickness, wherein the functional form is;

16. The method of claim 15 wherein the step of acquiring multiple pairs of image data comprises the step of simulating the intensities of the transmissive energy which passes through the calibration standard using one or more of the following simulation factors:

a) spectral characteristics of the source of transmissive energy; and

/or b) angular distribution of the source of transmissive energy; and

/or c) stopping power and spectral sensitivity of a transmissive energy detector; and

/or d) transmissive energy attenuation properties of the absorbing material as a function of energy/wavelength of the source of transmissive energy.

17. The method of claim 15 further comprising the steps of:

measuring foreground and background values for a combination of the first and second materials having unknown thicknesses; and

locating on the Background (y_b) vs. Delta Gray (Δ

G=y_f−

y_b) vs. First Material Thickness (t_M1) surface, background and Delta Gray image data values corresponding to the measured background and foreground values to determine at least one of the corresponding first and/or second material thicknesses.

18. The method of claim 15 further comprising the step of generating a Background (y_b) vs. Delta Gray (Δ

G=y_f−

y_b) vs. First Material Thickness (t_M1) and/or Second Material Thickness (t_M2) look up table using the fitted values for y₀, α

_i γ

_i and β

_i.

19. The method of claim 18 further comprising the steps of:

measuring foreground and background values for a combination of the first and second materials having unknown thicknesses; and

locating on the Background (y_b) vs. Delta Gray (Δ

G=y_f−

y_b) vs. First Material Thickness (t_M1) look up table, Background and Delta Gray intensity values corresponding to the measured background and foreground values to determine at least one of the corresponding first and/or second material thicknesses.

20. The method of claim 19 further comprising the step of interpolating between values in the lookup table.

21. A method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said method comprising the steps of:

providing a calibration standard for characterizing the imaging system wherein the calibration standard includes a first known thickness of the first absorbing material (denoted by t_M1,1) in combination with two different thicknesses of the second absorbing material (denoted by t_M2,1 and t_M2,2);

determining values of first and second foreground parameters (denoted by F₁ and F₂) wherein;

a) the first foreground parameter F₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,1; and

b) the second foreground parameter F₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,2;

determining values of first and second background parameters (denoted by B₁ and B₂) wherein;

a) the first background parameter B₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,1; and

b) the second background parameter B₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,2;

determining a first non-linear functional form, y₁(x), which describes values of foreground (y₁=F) as functions of the background (x=B) such that the first non-linear functional form;

a) approximates the previously determined values of the first and second foreground parameters (F₁ and F₂) in terms of the previously determined values of the first and second background parameters (B₁ and B₂);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate a third foreground parameter (F₃) at a corresponding third background parameter (B₃) to a reference background value (x=B_R), thereby determining a reference foreground value (y₁=F_R) at the reference background value (x=B_R); and

determining a second non-linear functional form, y₂(x), which describes reference foreground values (y₂=F_Ri) as a function of corresponding first absorbing material thicknesses (x=t_M1,i) such that the second non-linear functional form;

a) approximates a reference foreground value (y₂=F_R1) of the calibration standard first known thickness of the first absorbing material (t_M1,1) at the reference background value (x=B_R); and

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system.

22. The method of claim 21 wherein the step of determining a first non-linear functional form, y₁(x), further comprises the step of selecting hyperbolic functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system.

23. The method of claim 21 wherein the step of determining a second non-linear functional form, y₂(x), further comprises the step of inverting, either numerically or analytically, the second non-linear functional form to obtain a first material thickness (t_M1,K) corresponding to a given reference foreground value (y₂=F_RK).

24. The method of claim 21 wherein the step of determining a second non-linear functional form, y₂(x), further comprises the step of selecting a sum of exponential functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system.

25. The method of claim 21 wherein the steps of determining the values of the foreground and background parameters further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors:

a) spectral characteristics of an X-ray source; and

/or b) angular distribution of X-rays produced by the X-ray source; and

/or c) stopping power and spectral sensitivity of an X-ray detector; and

/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength.

26. A method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said method comprising the steps of:

providing a calibration standard for characterizing the imaging system wherein the calibration standard includes a first known thickness of the first absorbing material (denoted by t_M1,1) in combination with two different thicknesses of the second absorbing material (denoted by t_M2,1 and t_M2,2);

determining values of first and second foreground parameters (denoted by F₁ and F₂) wherein;

a) the first foreground parameter F₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,1; and

b) the second foreground parameter F₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,2;

determining values of first and second background parameters (denoted by B₁ and B₂) wherein;

a) the first background parameter B₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,1; and

b) the second background parameter B₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,2; and

determining a functional form of a non-linear function, y(x₁,x₂), which describes the value of the thickness of the first material (y=t_M1) as a function of the foreground and background (e.g., x₁=F, x₂=B) such that the non-linear functional form;

a) approximates a set of calibration data points {(t_M1,i,F_i,B_i)} containing the previously determined first material thicknesses (t_M1,i), foreground parameters (F_i) and background parameters (B_i);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters.

27. The method of claim 26 wherein the step of determining a functional form of the non-linear function, y(x₁,x₂), further comprises the step of selecting a sum of the product of two exponentials to represent the foreground parameters and a sum of single exponentials to represent the background parameters as the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system.

28. The method of claim 26 wherein the step of determining a functional form of the non-linear function, y(x₁,x₂), further comprises the step of inverting, either numerically or analytically, the non-linear functional form such that any one of y, x₁ or x₂ may be expressed as a function of the remaining two variables.

29. The method of claim 26 wherein the steps of determining the values of the foreground and background parameters further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors:

a) spectral characteristics of an X-ray source; and

/or b) angular distribution of X-rays produced by the X-ray source; and

/or c) stopping power and spectral sensitivity of an X-ray detector; and

/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength.

30. A method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said method comprising the steps of:

providing a calibration standard for characterizing the imaging system wherein the calibration standard includes first and second known thicknesses of the first absorbing material (denoted by t_M1,1 and t_M1,2) in combination with a thickness of the second absorbing material (denoted by t_M2,1 and t_M2,2);

determining values of first and second foreground parameters (denoted by F₁ and F₂) wherein;

a) the first foreground parameter F₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,1; and

b) the second foreground parameter F₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,2 in combination with the second absorbing material having the thickness t_M2,2;

determining values of first and second background parameters (denoted by B₁ and B₂) wherein;

a) the first background parameter B₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,1; and

b) the second background parameter B₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,2;

determining a first non-linear functional form, y₁(x), which describes values of foreground (y₁=F) as functions of the background (x=B) such that the first non-linear functional form;

a) approximates the previously determined values of the first and second foreground parameters (F₁ and F₂) in terms of the previously determined values of the first and second background parameters (B₁ and B₂);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate a third foreground parameter (F₃) at a corresponding third background parameter (B₃) to a reference background value (x=B_R), thereby determining a reference foreground value (y₁=F_R) at the reference background value (x=B_R); and

determining a second non-linear functional form, y₂(x), which describes reference foreground values (y₂=F_Ri) as a function of corresponding first absorbing material thicknesses (x=t_M1,i) such that the second non-linear functional form;

a) approximates a first reference foreground value (y₂=F_R1) of the calibration standard first known thickness of the first absorbing material (t_M1,1) at the reference background value (x=B_R) and a second reference foreground value (y₂=F_R2) of the calibration standard second known thickness of the first absorbing material (t_M1,2) at the reference background value (x=B_R); and

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system.

31. The method of claim 30 wherein the step of providing a calibration standard further comprises the step of selecting the second absorbing material such that the thickness t_M2,1 equals the thickness t_M2,2.

32. The method of claim 30 wherein the step of determining a first non-linear functional form, y₁(x), further comprises the step of selecting hyperbolic functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system.

33. The method of claim 30 wherein the step of determining a second non-linear functional form, y₂(x), further comprises the step of inverting, either numerically or analytically, the second non-linear functional form to obtain a first material thickness (t_M1,K) corresponding to a given reference foreground value (y₂=F_RK).

34. The method of claim 30 wherein the step of determining a second non-linear functional form, y₂(x), further comprises the step of selecting a sum of exponential functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system.

35. The method of claim 30 wherein the steps of determining the values of the foreground and background parameters further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors:

a) spectral characteristics of an X-ray source; and

/or b) angular distribution of X-rays produced by the X-ray source; and

/or c) stopping power and spectral sensitivity of an X-ray detector; and

/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength.

36. A method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said method comprising the steps of:

providing a calibration standard for characterizing the imaging system wherein the calibration standard includes first and second known thicknesses of the first absorbing material (denoted by t_M1,1 and t_M1,2) in combination with a thickness of the second absorbing material (denoted by t_M2,1 and t_M2,2);

determining values of first and second foreground parameters (denoted by F₁ and F₂) wherein;

a) the first foreground parameter F₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,1 in combination with the second absorbing material having the thickness t_M2,1; and

b) the second foreground parameter F₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t_M1,2 in combination with the second absorbing material having the thickness t_M2,2;

determining values of first and second background parameters (denoted by B₁ and B₂) wherein;

a) the first background parameter B₁ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,1; and

b) the second background parameter B₂ is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,2; and

determining a functional form of a non-linear function, y(x₁,x₂), which describes the values of the thickness of the first material (y=t_M1) as a function of the foreground and background (e.g., x₁=F, x₂=B) such that the non-linear functional form;

a) approximates a set of calibration data points {(t_M1,i,F_i,B_i)} containing the previously determined first material thicknesses (t_M1,i), foreground parameters (F_i) and background parameters (B_i);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters.

37. The method of claim 36 wherein the step of providing a calibration standard further comprises the step of selecting the second absorbing material such that the thickness t_M2,1 equals the thickness t_M2,2.

38. The method of claim 36 wherein the step of determining a functional form of the non-linear function, y(x₁,x₂), further comprises the step of selecting a sum of the product of two exponentials to represent the foreground parameters and a sum of single exponentials to represent the background parameters as the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system.

39. The method of claim 36 wherein the step of determining a functional form of the non-linear function, y(x₁,x₂), further comprises the step of inverting, either numerically or analytically, the non-linear functional form such that any one of y, x₁ or x₂ may be expressed as a function of the remaining two variables.

40. The method of claim 36 wherein the steps of determining the values of the foreground and background parameters further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors:

a) spectral characteristics of an X-ray source; and

/or b) angular distribution of X-rays produced by the X-ray source; and

/or c) stopping power and spectral sensitivity of an X-ray detector; and

/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength.

41. A method for calibrating an X-ray imaging system for quantitatively determining a first thickness, T_x, of an absorbing material in the presence of an additional, second thickness, T_y, of the absorbing material, where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the absorbing material, said method comprising the steps of:

providing a calibration standard for characterizing the imaging system wherein the calibration standard provides two known thicknesses T₁ and T₂ of the absorbing material;

determining values F₁ and F₂ reflective of transmitted X-ray beam intensities corresponding to transmission through thicknesses T₁ and T₂ of the absorbing material, respectively;

determining a functional form of an invertible, non-linear function y(x) which describes the variation of transmitted X-ray beam intensity as a function of thickness of the absorbing material;

determining values B and F reflective of transmitted X-ray beam intensities corresponding to transmission through the second thickness, T_y, of the absorbing material and through the combined thickness, T_x+T_y, of the absorbing material, respectively;

applying the previously determined functional form to determine T_y and T_x+T_y from the measured values of F and B; and

determining the unknown first thickness, T_x, as the difference (T_x+T_y)−

T_y.

42. The method of claim 41 wherein the step of determining a functional form which describes transmitted beam intensity as a function of thickness further comprises selecting a general functional form described by:

43. The method of claim 41 wherein the step of determining the values F₁ and F₂ comprises the step of simulating the transmitted intensities using one or more of the following simulation factors:

a) spectral characteristics of the incident X-ray beam; and

/or b) angular distribution of X-rays comprising the incident X-ray beam; and

/or c) stopping power and spectral sensitivity of an X-ray detector; and

/or d) X-ray attenuation properties of the absorbing material as a function of X-ray energy/wavelength.

44. An apparatus for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said apparatus comprising:

a calibration standard for characterizing the imaging system wherein the calibration standard includes at least one known thickness t_M1,i of the first absorbing material in combination with at least one thickness t_M2,i of the second absorbing material;

means for determining a value of foreground and background parameters (denoted by F and B) wherein;

a) the foreground parameter F is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having thickness t_M1,1 in combination with the second absorbing material having a thickness t_M2,i; and

b) the background parameter B is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,i; and

means for determining a non-linear functional form which describes values of the foreground and/or the background and/or the material thicknesses such that the non-linear functional form;

a) is consistent with the previously determined foreground parameter (F), background parameter (B), and thickness values;

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate the foreground and/or the background and/or the material thicknesses beyond the range of the calibration standard.

45. The apparatus of claim 44 wherein the means for determining a non-linear functional form further comprises:

means for determining a first non-linear functional form, y₁(x), which describes values of foreground (y₁=F) as functions of the background (x=B) such that the first non-linear functional form;

a) approximates the previously determined value of the foreground parameter (F) in terms of the previously determined value of the background parameter (B);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate a measured foreground parameter (F_M) corresponding to a first absorbing material having an unknown thickness t_M1,U in combination with a second absorbing material having a thickness t_M2,U to a reference background value (x=B_R), thereby determining a reference foreground value (y₁=F_R,U) at the reference background value (x=B_R); and

means for determining a second non-linear functional form, y₂(x), which describes reference foreground values (y₂=F_Ri) as a function of corresponding first absorbing material thicknesses (x=t_M1,i) such that the second non-linear functional form;

a) approximates a reference foreground value (y²=F_R1) of the calibration standard for the known thickness of the first absorbing material (t_M1,1) at the reference background value (x=B_R); and

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system.

46. The apparatus of claim 44 wherein the means for determining a non-linear functional form further comprises:

means for determining a functional form of a non-linear function, y(x₁,x₂), which describes the values of the thickness of the first material (y=t_M1) as a function of the foreground and background (e.g., x₁=F, x₂=B) such that the non-linear functional form;

a) approximates a set of calibration data points {(t_M1,i,F_i,B_i)} containing the previously determined first material thicknesses (t_M1,i), foreground parameters (F_i) and background parameters (B_i);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters.

47. A method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, said method comprising the steps of:

providing a calibration standard for characterizing the imaging system wherein the calibration standard includes at least one known thickness t_M1,i of the first absorbing material in combination with at least one thickness t_M2,i of the second absorbing material;

determining a value of foreground and background parameters (denoted by F and B) wherein;

a) the foreground parameter F is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having thickness t_M1,i in combination with the second absorbing material having a thickness t_M2,i; and

b) the background parameter B is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t_M2,i; and

determining a non-linear functional form which describes values of the foreground and/or the background and/or the material thicknesses such that the non-linear functional form;

a) is consistent with the previously determined foreground parameter (F), background parameter (B), and thickness values;

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate the foreground and/or the background and/or the material thicknesses beyond the range of the calibration standard.

48. The method of claim 47 wherein the step of determining a non-linear functional form further comprises the steps of:

determining a first non-linear functional form, y₁(x), which describes values of foreground (y₁=F) as functions of the background (x=B) such that the first non-linear functional form;

a) approximates the previously determined value of the foreground parameter (F) in terms of the previously determined value of the background parameter (B);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate a measured foreground parameter (F_M) corresponding to a first absorbing material having an unknown thickness t_M1,U in combination with a second absorbing material having a thickness t_M2,U to a reference background value (x=B_R), thereby determining a reference foreground value (y₁=F_R,U) at the reference background value (x=B_R); and

determining a second non-linear functional form, y₂(x), which describes reference foreground values (y₂=F_Ri) as a function of corresponding first absorbing material thicknesses (x=t_M1,i) such that the second non-linear functional form;

a) approximates a reference foreground value (y₂=F_R1) of the calibration standard for the known thickness of the first absorbing material (t_M1,1) at the reference background value (x=B_R); and

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system.

49. The method of claim 47 wherein the steps of determining the values of the foreground and background parameters further comprises the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors:

a) spectral characteristics of the X-ray beam; and

/or b) angular distribution of X-rays comprising the X-ray beam; and

/or c) stopping power and spectral sensitivity of an X-ray detector; and

/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength.

50. The method of claim 47 wherein the foreground parameters F_i are described by a general functional form, y_F:

51. The method of claim 47 wherein the step of determining a non-linear functional form comprises the step of selecting a function of the form:

52. The method of claim 51 further comprising the steps of:

selecting a reference background level (x=B_R);

determining the values of foreground minus background (F_Ri−

B_Ri) at the reference background level (B_R) for multiple known thicknesses of the calibration standard using the smoothly varying non-linear function y₁ which expresses the value of the foreground minus the background (y₁=F−

B) as a function of background (x=B); and

determining a second functional form y₂ which expresses the values of foreground minus background (F_Ri−

B_Ri) at the reference background level (B_R) for the multiple known thicknesses of the first absorbing material as a function of the thickness of the first absorbing material.

53. The method of claim 52 wherein the step of determining a second functional form y₂ further comprises the step of selecting a function which is a sum of exponentials of the form:

54. The method of claim 47 further comprising the step of producing a lookup table for values of (background) vs. (foreground minus background) vs. (thickness) for one or both of the first and/or second absorbing materials.

55. The method of claim 47 wherein the step of determining a non-linear functional form further comprises the step of:

determining a functional form of a non-linear function, y(x₁,x₂), which describes the values of the thickness of the first material (y=t_M1) as a function of the foreground and background (e.g., x₁=F, x₂=B) such that the non-linear functional form;

a) approximates a set of calibration data points {(t_M1,i,F_i,B_i)} containing the previously determined first material thicknesses (t_M1,i), foreground parameters (F_i) and background parameters (B_i);

b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and

c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters.